finding a sorting algorithm using genetic programming

Finding a Sorting AlgorithmUsing Genetic Programming

May 22, 2003Hyunkoo Jee

Defining a Virtual Machine and the Syntax of the Programming Language (1/4)

Resources that can be used by the VM•Array of registers : A[0], …, A[N-1]

• Index registers : I1, I2 (initial value : 0) Machine Instructions and the Syntax

•Syntax•One instruction (and its operands) per one line

•An instruction can have at most two operands

•Line number is assigned as 0, 1, 2, …


Example0: PUT I2 ← I11: INC I22: EXCH3: BR (I2 >= N-1) 54: BR (I1 < I2) 15: INC I16: BR (I1 < I2) 0


Instructions• EXCH : (Exchange) First, it compares A[I1] and A[I2]. And, if there is an inversion,

it exchanges the values of A[I1] and A[I2]. If “0<=I1<=N-1 and 0<=I2<=N-1” is not satisfied, this instruction doesn’t do anything.

• INC operand1 : (Increase) It increases the value of the register specified in operand1. Operand1 can be I1 or I2.

• DEC operand1 : (Decrease) It decreases the value of the register specified in operand1. Operand1 can be I1 or I2.

• PUT operand1 : It copies the value of a register into another register. Operand1 can be “I2 ← I1” or “I1 ← I2”.

• BR operand1 operand2 : (Conditional branch) If the condition specified in operand1 is true, jump to the line specified in operand2. operand1 can be…

• I1 <= 0• I2 <= 0• I1 >= N-1• I2 >= N-1• I1 < I2• I1 > I2• I1 = I2

operand2 is the line number to jump to.


A priori constraints that the programs must satisfy•Use the instruction “EXCH” exactly one

time

Defining the Cost Function (or Fitness Function)

Weighted sum of•Inversion

= #( {(i,j) | i < j and A[i] > A[j], 0<=i,j<=N-1} )

•Runtime= (how many instructions executed)

•Length= (how many lines)

Evaluation Implement the virtual machine, or an

emulating program Execute the individual candidates(=programs)

on this VM, and compute the cost function of each candidates.

Here we generate T arrays, each array has N elements. Let each individual sort these T arrays. And we compute the average of the T cost values of each individual.

If “runtime” exceed some threshold, stop the VM (unless it would run an infinite loop).

Defining the Evolutionary Operations 4 Mutation Methods

• Insert a new line•When inserting a line including “BR”, the “operand2”

should be in an adequate range.

•Don’t insert “EXCH” line.

• Delete a line•Don’t delete “EXCH” line.

• Swap two lines

• Mutate a line•Do not touch the instruction. Mutate the operands

only. To mutate an individual candidate, randomly

apply these 4 methods, O times.

Selection(how to generate the next generation)

Summary of the Procedure Make P initial individual candidates

• At first, each individual(=program) has only one line, i.e. “EXCH”• To generate the initial population, apply the mutation methods I times to each ind

ividual. Evaluate the cost function

• Generate T arrays, each of them has N elements.• Put these T arrays to each individual in current generation. And evaluate the cost

function f for each individual. Selection

• Classify the individuals in the current generation as : best group(S candidates), worst group(S), ordinary group(P-2S)(according to the cost-function ranking)

• Make the next generation with : best group, mutated best group, mutated ordinary group(Here we apply the mutation method O times, in order to mutate an individual)

Jump to “Evaluate the cost function” again, and repeat

Experimental Condition T = 20

• How many sample arrays, needed to evaluate the cost function of a candidate? N = 10

• How many elements in each array? I = 20

• How many times we apply the mutation methods, when we generate the initial population? O = 5

• How many times we apply the mutation methods, when we mutate an individual? Wruntime = 0.0005 Winversion = 0.9945 Wlength = 0.005 P = 500

• How many individuals in a generation? S = 120

• How many individuals in the best group (preserved for the next generation)? = How many individuals in the worst group?

Experimental Results

Experimental Results After the 2000th generation, the best individual doesn't change.

The best individual 0: DEC I2 1: BR (I2 <= 0) 12 2: PUT I1 ← I2 3: DEC I1 4: EXCH 5: BR (I1 <= 0) 0 6: DEC I1 7: BR (I1 < I2) 4 8: INC I1 9: PUT I2 ← I110: BR (I1 >= N-1) 311: INC I112: BR (I1 > I2) 8

• Runtime = 224, Inversion = 0, Length = 13• Cost Function = 0.177




I1 과 I2 를 N-1 까지 증가시키는 루프




I1 을 계속 감소시키면서 , A[I1] 과 A[I2] 가 inversion 된 경우 이 둘을 exchange

결국 A[0]~A[I2] 중 가장 큰 값이 A[I2] 로 들어가게 된다

Analysis of the Results selection sort (putting the correct value from right to left) Time complexity : O(N2) Not the optimal solution, but not so bad Diversity preserved It is important to define the virtual machine and its instructions ade

quately Future works

• Using crossover• Using simulated annealing• Using general x86 machine instruction• modular structure (make sub-routines and call them)• Recursive call → O(N logN) algorithm• Generalization → apply to other problems

Conclusion 주어진 문제를 해결하는 알고리즘을 찾기 위하여 , EA(Evolution

ary Algorithm) 를 이용하여 컴퓨터 프로그램을 진화시키는 전략 (Genetic Programming)

• 적절한 알고리즘을 찾아내기 어려운 문제가 주어진 경우 , 문제에 어느 정도 접근한 간단한 프로그램을 작성한 뒤에 , 이 프로그램을 진화시켜 볼 수 있다 . 그러면 문제를 더욱 잘 푸는 새로운 알고리즘을 찾을 수 있을 것이다 .

• 다항 시간 (polynomail time) 내에 푸는 알고리즘이 없는 NP class 나 그 밖의 문제들을 풀기 위해선 , 휴리스틱 (heuristic) 방법을 사용하여 어느 정도 타당한 해를 구하는 방법을 많이 쓴다 . 이때 사용하는 휴리스틱 알고리즘을 잘 선택하는 것이 성능에 큰 영향을 미친다 . 이런 경우 역시 EA 를 이용하여 더 좋은 휴리스틱 알고리즘을 찾아낼 수 있을 것이다 .

• 최적화 (optimization) 가 아닌 다양성 (diversity) 을 목적으로 EA 를 사용한다면 , 기대하지 못했던 새로운 기능을 하는 프로그램이 창발 (emergence) 할 수도 있을 것이다 .

finding a sorting algorithm using genetic programming

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